M2-brane surface operators and gauge theory dualities in Toda

A bstract We give a microscopic two dimensional N = (2, 2) gauge theory description of arbitrary M2-branes ending on N f M5-branes wrapping a punctured Riemann surface. These realize surface operators in four dimensional N = 2 field theories. We show that the expectation value of these surface operators on the sphere is captured by a Toda CFT correlation function in the presence of an additional degenerate vertex operator labelled by a representation ℛ of SU( N f ), which also labels M2-branes ending on M5-branes. We prove that symmetries of Toda CFT correlators provide a geometric realization of dualities between two dimensional gauge theories, including N = (2, 2) analogues of Seiberg and Kutasov-Schwimmer dualities. As a bonus, we find new explicit conformal blocks, braiding matrices, and fusion rules in Toda CFT.